Verified Answer :

  • Distribute \(\frac{2}{3}\) into \((9x - 15)\) to get \(6x - 10\).
  • Distribute the negative sign into \((-6x + 2)\) to get \(6x - 2\).
  • Combine the results \((6x - 10) + (6x - 2)\) to obtain \(12x - 12\).
  • The final simplified expression is \(\boxed{12x - 12}\).

Explanation

  1. Analysis of the Expression

The expression to simplify is \(\frac{2}{3}(9x - 15) - (-6x + 2)\). This expression consists of linear terms in \(x\) and constant terms. Our objective is to simplify this expression into the form \(Ax + B\) by distributing constants and combining like terms.

  1. Distributing and Simplifying Terms

First, we distribute the fraction \(\frac{2}{3}\) across the terms inside the first set of parentheses:
\[\frac{2}{3}(9x - 15) = \left(\frac{2}{3} \times 9x\right) - \left(\frac{2}{3} \times 15\right) = 6x - 10.\]
Next, we distribute the negative sign into the second set of parentheses \((-6x + 2)\):
\[-(-6x + 2) = -(-6x) - 2 = 6x - 2.\]
Combining these results, the original expression becomes:
\[(6x - 10) + (6x - 2).\]

  1. Combining Like Terms

Now we combine the like terms (the \(x\)-terms and the constant terms):
\[(6x + 6x) + (-10 - 2) = 12x - 12.\]
Thus, the simplified expression is \(12x - 12\).

Examples

Imagine you are calculating the total cost for a supply order where the base price depends on the number of units \(x\). If the first part of your order involves a discount and the second part involves a tax or a surcharge, you might write an expression like this. Simplifying it helps you quickly determine the final formula for your cost calculation, making it easier to predict expenses for any number of units ordered.

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